Suppose a species has limitless resources and the ability to construct megastructures of an astronomical scale.

Would it be possible, given antigravitational technology (hypothetical), to create a planet with the diameter of a light year? Suppose there is an internal structure inside the planet that enables the gravity of the planet to be cancelled at correct points to ensure stability. This would stop the planet from collapsing in on itself?

Perhaps without antigravitational technology the planet would turn into a blackhole due to the massive mass (assuming relative composition to that of the earth)?

$\begingroup$You, my friend, have created a black hole. Unless you have antigravity technology.$\endgroup$
– Xandar The ZenonMar 14 '16 at 13:48

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$\begingroup$What you describe here, given that it's empty inside, is essentially a Dyson's Sphere with the only difference being that you imagine the creators want to live on the outside, not on the inside of the structure. en.wikipedia.org/wiki/Dyson_sphere$\endgroup$
– makingthematrixMar 14 '16 at 13:52

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$\begingroup$I agree with the other answers here that it's technically possible - especially with the inclusion of antigravity and such (at that point you can do pretty much whatever you want). But if I were to see this in a story, I think my first reaction would be, "But WHY?" I can't even IMAGINE what problem this advanced society would be trying to solve by creating a "planet" 1 light-year across! It's not like they would be sitting around saying, "Heh, wouldn't it be awesome if..." So it's an interesting idea, but I think a big part of this would be making sure you have a good "why" to go along with it$\endgroup$
– loneboatMar 14 '16 at 20:54

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$\begingroup$I agree that this idea could lack motivation. It it just to say to the readers that it's a very big planet? This would be like 742 millions Earths, way bigger than Niven Rings or Dyson's Spheres. Wouldn't you need an even bigger sun to illuminate its surface? Antigravitational handwaving is just one of the things you would need to make your world credible: what about cosmology, rotation, light behavior outside the planet, tides, atmosphere, seasons, geology, internal heating, volcanos... ?$\endgroup$
– ChaoticMar 15 '16 at 17:28

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$\begingroup$It may not become a black hole as calculated below, but how would you prevent a rogue black hole collision? In fact, what would a rogue black hole even do? Localized destruction that is repairable...? Guaranteed complete structural failure...? Doesn't the probability of rogue black hole collision or hell a gamma ray burst headshot go up by considerable orders of magnitude at this scale?$\endgroup$
– FriksterMar 15 '16 at 19:36

13 Answers
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As mentioned, anti-gravity and or gravity generators will let you pull magic out of thin air, but let's try it without magic. I can't imagine why anyone would bother doing it, but I decided to see what happens and ended up with a totally insane project that uses a lot of carbon nanotube (CNT) and crazy amounts of energy. But, you don't need unproven technologies like anti-gravity or unobtanium. $\ddot\smile$

Creating the Beast

The Schwarzschild radius of a black hole is $r_s<{2MG\over c^2}$. Solving that for mass, given a ½ ly radius, we get $M<3.19\cdot 10^{42}kg$ to avoid being a black hole.

The surface area of a sphere is $A=4\pi r^2$. The volume of spherical shell is $V\approx A\Delta r$ for $r\gg\Delta r$. The mass of the shell is $M=V\rho\approx A\Delta r\rho=4\pi r^2\Delta r\rho$. Solving that for thickness, given ½ ly radius, the mass given above, and the density of CNT (about $1.3 {g\over cm^3}$), we get $\Delta r<8727 km$.

The compressive strength of CNT is around 416 MPa. The inside of the sphere feels the pressure of the entire mass weighing down on it. The force is given by $f=Ma$, where $a$ is acceleration at the surface. $a={MG\over r^2}$, so $f={M^2G\over r^2}$. Pressure is $p={f\over A}={{M^2G\over r^2}\over 4\pi r^2}={M^2G\over 4\pi r^4}$. Rearranging gives $M=\sqrt{4p\pi r^4\over G}=r^2\sqrt{4p\pi\over G}$. Solving for mass, given 416 MPa and ½ ly radius, we get $1.98\cdot 10^{41}kg$, which is not a black hole, yay.

Using that mass, we can calculate a thickness of 55 km, surface acceleration of 0.0061 g (about $1\over 164$ Earth gravity), not a black hole, and the CNT construction can withstand the pressures. Of course, getting 10 billion solar masses of carbon nanotubes is a bit of a feat, but it's not outside the realm of just-possible.

You wouldn't be able to cover the entire thing with people, dirt, etc., and you'd need some source of energy (maybe a Dyson sphere surrounding a super-massive star in the center of your CNT planet?), but it's doable without magic.

Add some "Gravity"

As Zsolt Szilagy points out, pretty much any kind of rotation you can actually notice is going to wreak havoc with shear forces, but you might be able to spin it just fast enough to get some normal Earth gravity along the equator if you put your people on the inside. Wikipedia says the breaking length of CNT is around 4700 km under 1 g, so the CNT should stay together while being flung outward at 1 g. Centripetal acceleration is given by $a={v^2\over r}$ for uniform circular motion. Solving for speed, given 1 g and ½ ly radius, we get 0.718 c at the equator. Not impossible, but it's going to take a long time to get there.

Notes

Also, angular kinetic energy is $E={1\over 2}I\omega^2$, and $I={2\over 3}Mr^2$ for a thin, spherical shell. $\omega={v\over r}$, so $E={1\over 2}{2\over 3}Mr^2({v_\text{equator}\over r})^2={1\over 3}Mv_\text{eq}^2$. Solving for energy, given mass of $2\cdot 10^{41}kg$ and $0.718c$ speed, we get $3\cdot 10^{57}J$, which means converting about 8% of the Milky Way's mass to kinetic energy over some insanely long time period to do it.

Of note, if you're going to spin the "planet" and put people on the inside, you might as well save a bunch of material and just make it a Ringworld-style ring (it's still going to be insane though). Also, at 0.7 c, we're getting into relativistic territory, so the Newtonian equations aren't perfect, but I don't think it's anything sufficiently advanced scientists can't handle if they made it to this point. I'd be more worried about extrapolating CNT strengths from $\mu m$ scales to $km$ scales. And rogue stars.

$\begingroup$@LorenPechtel: There's a lot of mass, but it's spread really thinly (half the density of Earth's crust). The relativistic surface gravity equation is $\sqrt{r\over r-r_s}{MG\over r^2}$, where the root is a "correction factor" and $MG\over r^2$ is the normal calculation. $r_s\propto M$, so if M is 6.25% of the mass required for a black hole, then $r_s$ is 6.25% of current radius. This gives a correction factor of $\sqrt{1\over 0.9375}=1.033$. So my calculation for pressure is about 3% too low -- a lot smaller than the error for assuming CNT strengths scale perfectly.$\endgroup$
– MichaelSMar 15 '16 at 6:14

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$\begingroup$My thought process is like this: the outer-most layer of the shell (say 1mm thick) weighs almost nothing. The next layer is twice the weight. Etc. The inner-most layer has the weight of all the other layers plus its own weight -- that's the layer that's going to break first. There's around $10^{17}$ Earth masses weighing down, but it's at $10^{-2}g$ and spread over about $10^{17}$ times the area, so the pressure is pretty comparable to pressures near the surface of Earth. Not nearly enough for degenerate matter to form.$\endgroup$
– MichaelSMar 15 '16 at 6:22

$\begingroup$The dominating stress in the shell will be compression parallel to the surface, which your calculation doesn't seem to take into effect. The lateral compression force needed to keep a spherical dome up is proportional to the weight of the shell per unit area, but also proportional to the radius of the sphere. And a radius of half a light year will by far dominate over the fact that the shell weighs only 1/300 of what it would in standard gravity.$\endgroup$
– hmakholm left over MonicaMar 15 '16 at 15:18

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$\begingroup$I wonder if Kcronix is clear that when you talk about building a "shell", that means it's hollow--in a comment Kcronix had said "What I'm proposing is not empty inside". If you wanted a non-hollow sphere of this type, then since the radius is 0.5 light years = 4.73 * 10^15 meters, and the volume is (4/3)*pi*r^3 = 4.43 * 10^47 cubic meters, then with a mass just under 3.19 * 10^42 kg, the average density would be just under 0.72 * 10^-6 kg/m^3 = 0.72 * 10^-9 g/cm^3, similar to our atmosphere's density between 80 to 100 km up according to this.$\endgroup$
– HypnosiflMar 15 '16 at 21:32

$\begingroup$@MichaelS: No, it's the same for a free-body diagram for small spherical cap of your self-gravitating shell. Suppose the radius of the sphere is $R$ and of that of the cap is $r$, the gravity on the shell material is $W$ per unit area, and the lateral compression force is $F$ per unit length of a cut. Then the gravity on the cap is $Wr^2\pi$, which has to balance with the outwards component of the force in the cap's edges, which is $F\cdot 2\pi r\frac{r}{R}$ (in which $\frac rR$ is the fraction of the lateral force at the edges that actually goes in the direction of the weight). ...$\endgroup$
– hmakholm left over MonicaMar 16 '16 at 8:27

Given anti-gravity and a big enough power source, yes, a planet like that is possible But, really, by introducing anti-gravity, an author can do pretty much whatever they want as long as the universe rules stay consistent.

Without anti-gravity, the amount of mass required to make a 1 light year sphere out of iron would have long ago turned into a very large black hole.

As a fun little addendum, let's look at the surface gravity of this planet. Earth's surface gravity is $1~\text{g}=9.8~\frac{\text{m}}{\text{s}^{2}}$. The density of iron is $7874~\frac{\text{kg}}{\text{m}^{3}}$.

$\begingroup$Interesting. I have heard of stars many times larger than 3 solar masses that are not black holes however, are these the correct figures? Tho i suppose the flow thingy from the inside to the outside might sort it out (cant remember what that radiation pressure is called)$\endgroup$
– RugnirMar 14 '16 at 14:13

$\begingroup$@Green Not just total mass, total mass within a radius. Igniting fusion will provide enough pressure to counteract the force of gravity in the core to some extent, keeping the density below the threshold required for black hole formation. I think any element lighter than iron will ignite fusion prior to collapsing into either a neutron star or a black hole. Eventually, it will fuse up to iron (if massive enough), go supernova, and spit out a black hole.$\endgroup$
– ckerschMar 14 '16 at 15:11

I am not making fun of you but given the tremendous ability to manipulate their environment demonstrated by constructing this thing (the ability to move the mass of many solar systems, redirect the energy of many stars to illuminate it, and generally technology indistinguishable from magic), doing all this this no problem. But if they were able to harness these vast capabilities to make your structure, what sort of problems might a race like this encounter? Would anything be a "problem" to them at this point?

When you add anti gravity and gravity generating technology to hold everything in place and manipulation of vast amounts of energy to provide heat, your race may not be bound by the law of conservation of energy too.

So my question is, "Why do you want to put such a thing in you story?" Putting such a fantastic thing in your story should serve a narrative purpose. If we knew that purpose we may be able to answer your question better.

"If your race has the ability to create such a thing, what would such a race not be able to do?".

$\begingroup$To question the conditions set by the person that posted the question is a thing for comments, which you would be able to post when you have earned a little more reputation. Currently, this does not answer the question.$\endgroup$
– SE - stop firing the good guysMar 14 '16 at 16:31

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$\begingroup$I disagree with @Hohmannfan and fi12. I think this is a very reasonable answer to the question. Despite the question's extreme conditions, the asker is still asking "would it be possible", and this answer rightly points out that there are many levels of "no" beyond the few-yet-extremely-extreme ones already addressed. And that logically leads to the rest of the answer, which I would say is very on-point. The asker is asking for what they might not have considered, and this points in that direction, I would say better than most of the other answers do.$\endgroup$
– DronzMar 14 '16 at 18:46

You probably need to make it empty inside (like a ball) to avoid immediate gravitational collapse. Also, this makes obtaining the required amount of building material more realistic.

A body so large still would require some very advanced technologies to use, but may be manageable then.

You could then use some strong light source in the center that would keep the planet blown up like a huge solar sail. But this should be some special technology; no solar sail would work in such a distance from the ordinary star.

The density needed to become a black hole goes down as the radius of object increases. For the planet that you're talking about, if it had a density of $0.00000001488~\frac{\text{g}}{\text{cm}^{3}}$, it would collapse into a black hole. The density of the Earth is around $5.51~\frac{\text{g}}{\text{cm}^{3}}$.

You would need this planet to be something like a shell of unobtanium, or a lot of antigravity devices to stop from collapsing.

$\begingroup$Whether an object collapses due to its gravity depends on what’s going on inside. E.g., a star doesn’t collapse due to its nuclear fusion inside, but will collapse once the nuclear fusion stops, despite the density hasn’t changed. So how can you make such a statement based on the density only, without knowing anything else about the object? And where does that number come from?$\endgroup$
– HolgerMar 15 '16 at 11:23

$\begingroup$@Holger Whether an object collapses due to it's own gravity, and whether it turns into a black hole, are two different things (although a black hole will always collapse). You're right that the internal pressure gradient will stop a star from collapse, but IF you were to compress that star past it's Schwarzschild radius, spacetime is warped so much that no other forces can matter. A given radius can give you the upper bound on mass, and therefore density.$\endgroup$
– LacklubMar 15 '16 at 12:14

$\begingroup$Oh, I see. So the problem here is the density looking ridiculously low while the actually allowed mass still is insanely high considering the volume of the object. So all it says, is, there have to be some hollow areas inside that object (unless handwavy antigravity comes into play) to compensate…$\endgroup$
– HolgerMar 15 '16 at 17:48

$\begingroup$@Holger Exactly. Either there is going to be antigravity or unobtanium. Nothing else physically works$\endgroup$
– LacklubMar 15 '16 at 17:57

As the others have said, without anti-gravity it won't work, with anti-gravity you can pretty much do whatever you want.

Just a few things to think about

With a diameter of 1 light year ($5.879 \cdot 10^{12}$ miles) you get a circumference of $1.85 \cdot 10^{13}$ miles, and a surface area of $1.09 \cdot 10^{26}$ square miles.

The Earth only has a surface area of $1.969 \cdot 10^{8}$ square miles, so this thing is going to be really really big. If you took a big bag of Earths and skinned them like oranges, you would need $5.535805 \cdot 10^{17}$ Earths to cover this monster.

The point is that heating and lighting the place is going to be hard. You'd essentially want fleets of Sun-sized stars orbiting the planet in order to keep things warm and to provide enough light to grow things.

If you only put them around the equator, one AU away from the surface and spaced them 2 AUs apart, you'd need $99,505$ suns to circle the planet. However you'd need more than one band to keep the place warm, so maybe 2 bands at the 45th parallels too...

Just to put the numbers in normal notation instead of scientific notation:

surface area of $1.09 \cdot 10^{26} = 109,000,000,000,000,000,000,000,000$ square miles

$5.535805 \cdot 10^{17} = 553,580,500,000,000,000$ times the surface area of Earth

$\begingroup$You'll need a lot more stars. If one band is at the equator and the next is at 45 degrees there will be a point 500 times as far as pluto from the nearest star.$\endgroup$
– Loren PechtelMar 15 '16 at 5:03

$\begingroup$@LorenPechtel Yeah, I kinda thought I might, but I wasn't sure what it would be like to have that many stars spaced only a couple AU from each other; if the heat might multiply a lot more. I suppose it could end up looking like the latitude lines on a globe, spaced out every 10 degrees.$\endgroup$
– AndyD273Mar 15 '16 at 14:35

$\begingroup$Note that a galaxy has roughly 2e17 Earth-masses in it (total). So if the thickness is 1% of Earth, this weighs roughly 1/100th of an entire Galaxy.$\endgroup$
– YakkMar 15 '16 at 17:41

So long as you abandon the notion of "planet" and will accept a sphere of the requisite size this could be built although I can't imagine it being done by anything less than a K3 civilization.

As others have shown a solid mass is out of the question. A thin shell is the only possible way to do this--which leaves the problem of how to support it. Fortunately, we can do this without any handwavium.

The inside surface of the sphere is a huge collection of maglev tracks. Each track is occupied by a super-train--they are 6.28 light years long, the head coupled to the tail. They are moving far above orbital velocity and thus exert an outward force. Enough trains going fast enough and you can support your sphere. (We can't calculate the speed without knowing the ratio between the mass of the trains and the mass of the shell. I would not be surprised to learn that the speed needs to be relativistic.)

$\begingroup$+1 because using trains to cancel inertial force would be awesome for the 10 minutes or so it would work (before harmonics caused the entire shell to break apart and then collapse-in on itself from the gravity-well created by 6.3-Light-Year-Long trains de-railing)$\endgroup$
– Signal15Mar 17 '16 at 14:18

$\begingroup$@Signal15 You think a species that could build something like this couldn't keep it from being shaken apart by harmonic issues? And why would there be harmonic issues anyway? I described them as trains but it's really just spinning rings on a maglev track.$\endgroup$
– Loren PechtelMar 17 '16 at 20:54

Once you cancelled gravity and made sure you have enough energy sources, tectonics might become an issue. Make sure that planet does not rotate (like almost every known planet does). The centrifugal forces would need a year to cascade through - without beeing able to provide the math, I could imagine the planet would rip apart when the surface rotates at considerable fractions of the speed of light, or when parts of it have a differing momentum.

$\begingroup$Wouldn't centrifugal forces propagate with the speed of sound in the material, not the speed of light in vacuum?$\endgroup$
– a CVnMar 14 '16 at 14:48

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$\begingroup$@MichaelKjörling Yes, to an upper limit of c. I did some maths on a prior answer (about systematically deconstructing Earth to build a bridge to Alpha Centauri)[worldbuilding.stackexchange.com/questions/34279/…. Only ended up needing a speed of push (sound) at a tiny fraction of c at the time, but it was high enough to be measured in fractions of c. Further research lead me to a proof showing that a neutron star can't exceed 3.2 solar masses or its speed-of-sound-in-the-star-material would exceed c$\endgroup$
– Draco18s no longer trusts SEMar 14 '16 at 18:24

If in theory somehow something were possible to make a sphere 1 LY across made of something that somehow has a stable anti-gravity effect over such, then there are still many other basic conflicts with reality to consider, such as:

Have you really considered how large 1 LY is in comparison to any other known physical object in the universe, and in comparison to Earth and other planets in the solar system?

What is the thickness of the crust and the density of the interior that you have in mind? Any answer for this that even involves an earth-like surface over that distance, I would expect to add up to more matter than exists in multiple galaxies. What process is going to gather, transform and assemble your planet?

Given the scale of effort and ability needed to build such a thing, why would anything that advanced and powerful ever choose to use all that power to do that?

There are many ways in which the thing you are talking about is not going to be a planet, for many things we commonly think of as a planet.

For example, it won't rotate. Rotating something that large at any noticeable speed is practically impossible in a variety of ways due to several considerations from physics. For example, it will involve ridiculous amounts of shear force that would rip it apart. For another example, the resulting speeds at the surface would be relativistic, since the planetary circumference is 3.14159 Light Years - so there would be crazy time dilation effects between latitudes, and even if somehow the equator could go near the speed of light, one day would be over three years long. (Rotation relative to the direction of a bright light source is what provides days to planets.)

Also, things won't orbit around it (and it won't orbit around things) like Earth's moon and sun do, because it would take ridiculous amount of time for them go around the planet or vice versa, and they'd not be at scale with the planet or else they'd me even more ridiculous and impossible than the planet itself is. Even if you wanted a moon to go around as fast as possible, skimming near the surface of the outer atmosphere at nearly the speed of light (maybe you can project all that gravity onto the moon?), it's still going to take over three years to make one orbit, and the time visible overhead is going to be a small fraction of that. The geometry alone is off by a huge amount.

Given that you probably have no sun, what's going to provide surface heat and light?

As AndyD273 pointed out, you'd need something like "fleets of suns" to keep the surface heated, but again those multiply the impossibility of the construction project, and then there's the impossible task of getting them arranged in some sort of movement pattern that somehow regularly covers a sphere without problems. There probably exists no orbital pattern that would work, so you probably need the ability to control the movements of an astronomical number of suns all the time.

Not even space is a perfect vacuum. There are forces, gravity included, pushing, pulling, oscillating, etc., and the larger an object in space, the more these forces will act upon it.

Imagine a mile wide ball of leaves. Imagine said ball floating around our atmosphere about three miles up. Now, imagine the inconsistent wind blowing half the ball from the north and half the ball in the south... blowing twice as fast on the top of the ball as on the bottom. Imagine the flocks of birds, airplanes and debris from space colliding with it. Imagine the clouds passing through, oftentimes heavy with precipitation, oftentimes thunderous and lightning...ous... and then, finally, there's the gravity. All these forces acting on your ball of leaves would tear it apart, and the larger the ball, the more area it occupies, the more forces will act on it. The wind alone would be coming from so many diverse directions at so many diverse speeds that it would break your leaf ball into a million pieces. This is true for our atmosphere, for our ocean and, yes, even for the vacuum of space.

We know that the sun, moon and even the other planets have a significant-to-negligible gravitational pull on our planet. A light year wide planet would be subjected to the pull of who knows how many stars... including the one megastar or collection of lesser stars used to heat it. Speaking of heating it, what is the molten core situation for a planet like this? It seems there's only three options: virtually no molten core whatsoever when you get down to it, a thin crust of rock built on to of a sun (game over) or some magical, potentially impossible sweetspot that gives you a very large molten core that is both hot enough to stay molten while heating your planet AND not so hot that... well... one could only imagine. Hmh...

Long story short, the science and power needed to do something like this are so great that you'd have to be almost literally capable of anything in order to make it work. You'd have to be at the point where not only would creating the planet be no sweat, but handling every single problem that arises would be equally easy to solve, because you're like some kinda god or something. On that level, science would be irrelevant. It wouldn't be a matter of possible... it would be a matter of whether you wanted to or not.

Assuming basic science fails us (and it does), the BFP would have to be made one of several ways.
1.) Grown from a matrix that scavenges raw materials as it grows, extruding new material the same way a baby growing develops new skin. The "epidermis" would change structures at each stage to best support the newest pressures. The surface would have localized energy ports (self replicating) for harvesting new material from the surrounding system and be able to communicate shifts in spin to maintain the shape while still "flexible". Upon near completion all stored / harvested energy will have been distributed among all the sections to maintain balance and then the radiators would be launched on tethers as the rotation was slowed and the surface hardened. The entire innards of the sphere would be laced with slender ultralight cables much like a 3D bicycle wheel spoke system with the ends of the cables forming a sphere deep within the BFP (well 1/2 ly). These cables are only really needed to communicate and monitor stresses at first.

Once the radiators are launched (from every other section) they will serve multiple purposes. First, a specific radiation will be generated from the surface of each section using the matter harvesting generators and this will end the growth phase and start the unification phase. Each section will bond to its neighbor and use interlocking tendrils to help bind each plate to a perfect mating. Whatever the final geometric shape would be (boggles the mind) each section and it's immediate neighbors would be essentially a planetary surface and this is where the generators/radiators come into play.

The generators (the every other ones) would generate a beam of energy that struck the radiator spheres and those would release it as the most desirable visible frequency and a healthy modicum of infrared in alternating day night cycles tuned to each neighbor. The crust of the BFP would be roughly 12,750 km providing enough mass at every point to simulate 1g and provide structural integrity to the entire sphere, with a 500 m Earth like layer of organic topsoil. The rest of the 12,250 km would be layers of support structure, global and local supply/storage and automated repair facilities. This sectional mass would be insignificant to the size of the final construct, and too far away from other masses to cause problems. The side to side attraction of the sections would do little more than further strengthen integrity.

The harvesting generators not used for lighting would be responsible for gathering new matter and energy as needed and everything from asteroids to solar systems would be munched into what ever the BFP required, cutting swaths in whatever it encountered.

In this fashion, it would be 100% self sufficient and capable of travel by use of the same harvester energy beams used to power gravimetric engines. By no means could this ever be called a planet, it is more of a living machine.

2.) Genetically engineer some sort of bio-mechanical robotic creature(s) to build it, then consume it building on the outside of the existing structure and grow each cycle leaving your with a perfect honeycomb sphere made of whatever material you program it for and whatever depth is desired for best structure strength. You would also need to incorporate the same sort of energy harvesting beam technology.

In both cases, a holographic communication system would be used to defer delays in signaling, the light being transmitted via the fibers from each section and the data encoded in the holographic standing wave. The light might take 1 year to get from edge to center to all edges, but the standing wave should be affected instantaneously (in theory). Much like if you threaded a single glass rod from Chicago to Australia and tapped it, normal wave propagation comes into effect for the "tap" to be heard at the other end, but if you pull or push on the end, the other end moves at the same instant, no propagation needed.

$\begingroup$Your standing-wave and glass rod statements are wrong. If you push or pull a rod, the other end doesn't move until the force is transmitted through it at the rather slow speed of sound.$\endgroup$
– JDługoszJun 4 '16 at 16:51

Yes. And lighting it is possible without unobtainium. Sunlight is 1000W/m^2.
Over a billion seconds (30 years) you will need 10^12J. Direct mass to energy transformations give 10^17J/kg so that is 10^17/10^12 x30 years =3,000,000 years per kilo per meter squared. So if it has enough mass per area to give earth gravity at 1.2*10^10 kg/m^2 then it can be lit for at most 1.2*10^10*3000000=3.6*10^16 years (earth years) (actually twice as much because of night) before all the mass has turned to energy.
I would recommend the maglev track idea with large tanks of matter(any) underground. These are slowly fed to mini black holes whose hawking radiation is beamed up out a hole in the ground, bounces off a mirror kept aloft by the force of the light and illuminates the ground.